Developing 3D Anisotropic Mechanics Developing 3D Anisotropic - - PowerPoint PPT Presentation
Developing 3D Anisotropic Mechanics Developing 3D Anisotropic Mechanics Model of Powder Compaction Model of Powder Compaction Wenhai Wang Advisor: Dr. Antonios Zavaliangos Department of Materials Science & Engineering 12-10-2004 1
1
2
Powder compaction Literature review
Phenomenological models Introduction of VUMAT Results and discussion
Anisotropy in powder compaction Anisotropic models
3
Metal Industry Metal Industry Pharmaceutical Industry Pharmaceutical Industry Food Industry Food Industry Chemical Industry Chemical Industry Ceramics Industry Ceramics Industry
4
How do we get there? How the product performs?
5
10 mm 50 µm
Meso Meso-
scopic Macroscopic Macroscopic Microscopic Microscopic
Network Network Models Models Micromechanical Micromechanical Models Models
Phenomenological Phenomenological Models Models
MPFEM MPFEM Atomistic Atomistic Simulation Simulation
6
References: 19-20
Look into microscopic level, the local anisotropy is considered and macro- behavior is deduced.
Microscopic
References: 11-18
Study the particle collection. (statistics information are inherently considered)
Meso-scopic
References: 1-10
The powder is considered as a continuum.
Macroscopic
1. H.A. Kuhn, C.L. Downey, Int. J. Powder Metall. 7 (1) (1971) 15-25 2. R.J.Green, Int. J. Mech. Sci. 14 (1972) 215-224 3.
4. D.C. Drucker, W. Prager Q. Appl. Math. 10 (1952) 157-175 5. A.N. Schofield, C.P. Wroth, McGrawHill, London, 1968 6. F.L. DiMaggio, I.S. Sandler, J. Eng. Mech. Div., Proc. – ASCE 96 (1971) 935-950 7. PM Modnet Computer Modelling Group, Powder Metall. 42 (1999) 301- 311 8. I.C. Sinka, J.C. Cunningham, A. Zavaliangos, Powder Tech. 133 (2003) 33-43 9. Sofronis P, Memeeking RM, Mechanics of Materials 18 (1): 55-68 May 1994 10. A, Zavaliangos L, Anand J. of the Mech. and Phy. Of solid 41 (6): 1087- 1118 JUN 1993
Selected References:
2-15
Solids 47 (1999) 785-815 14. M.Kailasam, N. Aravas, P. Ponte Castaneda CMES, Vol. 1, pp. 105-118 2000 15.
16. P.R. Heyliger & R. M. McMeeking, J. Mech. Phy. Of Solid 49 (2001) 2031-2054 17.
18. C.L. Martin, D. Bouvard, Acta Mate. 51 (2003) 373–386 19. Francisco X. –Castilloa S. and Anwarb J., Heyes D.M. J. of
7
Yield is pressure dependant Single state variable – Relative Density Model parameters can be calibrated by experiments They can be implemented in FEM to simulate complex shape compaction operations.
1 ) ( ) ( ) , , (
2 2
= − + = Φ p D B D A D p σ σ
equivalent stress P hydrostatic pressure D relative density
Ellipse Model
tensile compressive
p
Relative density increase
8
= Experimental Measurements
“Kuhn-Shima” model (1970’s)
9
20 40 60 80 100 120 140 160 180 20 40 60 80 100 120 140 160 180
Hydrostatic Stress, MPa Effective Stress, MPa
10
I.C. Sinka, J.C. Cunningham and A. Zavaliangos Powder Technology 133 (2003) 33– 43 PM Modnet Computer Modeling Group, Powder Metallurgy, Vol. 42, 1999, 301-311
11
Apply Drucker-Prager Cap model (DPC) into ABAQUS/Standard simulation (All parameters are taken as function of RD); Model predicts the inversion of radial variation of relative density and hardness (lubricated V.S. unlubricated die).
I.C. Sinka, J.C. Cunningham and A. Zavaliangos Powder Technology 133 (2003) 33– 43
12
Current DPC model in ABAQUS/Standard is OK but convergence is a problem. ABAQUS/Explicit does not have flexible enough DPC model but it can address more complex geometry problems. To this end, a versatile version DPC model (All parameters are taken as function of RD) was implemented in VUMAT of ABAQUS/Explicit.
ABAQUS
Integrating
) ( ), ( ), ( t F t V t X
i i i
) ( t t X i ∆ +
i
ε ∆
) ( t t
i
∆ + σ VUMAT
Solving equations
) (t
i
σ
) ( t t Fi ∆ +
13
Simple compression Constraint compression Hydrostatic compression Simple tensile Constraint tensile Hydrostatic tensile
Loading conditions:
Porosity
0.69 0.7 0.71 0.72 0.73 0.74 0.75 0.01 0.02 0.03 0.04 0.05 0.06 Time (s)
s11/s22
20000 40000 60000 80000 100000 120000 140000 0.01 0.02 0.03 0.04 0.05 0.06 Time (s)
Material: Avicel
ABAQUS /Standard VUMAT
14 Simple compression Constraint compression Hydrostatic compression Simple tensile Constraint tensile Hydrostatic tensile
Loading conditions:
S22
20000 0.05 0.1 0.15 0.2 Time (s)
Porosity
0.69 0.71 0.73 0.75 0.05 0.1 0.15 0.2 Time (s)
S11
10000 20000 30000 40000 50000 60000 70000 80000 90000 0.05 0.1 0.15 0.2 Time (s)
Material: Avicel
ABAQUS /Standard VUMAT
15
Simple compression Constraint compression Hydrostatic compression Simple tensile Constraint tensile Hydrostatic tensile
Loading conditions:
Porosity
0.62 0.64 0.66 0.68 0.7
Strain
S11
Strain
S22
Strain
The origin of the difference is the Elastic modulus. It appears that ABAQUS/Standard does not update the modulus. Simulations with higher modulus show no difference. ABAQUS /Standard VUMAT
16
0.0125 Porosity
Unlubricated Lubricated ABAQUS/Explicit results (run with VUMAT) show good agreement with experimental results!
0.25 0.30 0.35 0.40 0.45 0.50 0.002 0.004 0.006 0.008 0.01 0.012 0.014
radius porosity
0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.002 0.004 0.006 0.008 0.01 0.012 0.014
radius porosity
17
Density distribution is well predicted!
How about the strength & modes of fracture prediction?
Shear failure Region Cap Region
Non-associated plasticity Associated plasticity Failure+Dilation Densification
DPC model shows the different densification trend when the stress hit different yield surface regions. (Shear failure region v.s. Cap region)
18
Diametrical Compaction Tablets compacted with different die lubrication show different fracture behaviors. Die Compaction
Unlubricated Lubricated
Diametrical compression tests are carried out in the pharmaceutical industry to test the “hardness” of tablets.
19
Final Relative density distribution (2-D) Initial Relative density distribution (3-D)
Die Compaction Diametrical Compaction Mapping Mapping
Unlubricated Lubricated
20
Low density in the middle somewhat indicates the initial fracture development from the center.
Before failure After failure
21
Convergence problems may happen when larger time step was selected.
Before failure After failure
22
Force Displacement
20 40 60 80 100 120 140 0.00 0.20 0.40 0.60 0.80 1.00 1.20
Displacement (mm) Force (N)
Unlubricated Lubricated
20 40 60 80 100 120 140 160 180 200 0.2 0.4 0.6 0.8 1 1.2 Distance, mm Force, N
lubricated die unlubricated die 0.374 0.380 0.416 0.464 0.505 0.560 0.590 0.422 0.433 0.472 0.510 0.559 0.612
Simulation Experiment
0.59 0.59
23
Die Compaction Isostatic Compaction Triaxial Compaction Σ Τ Σ Τ Σ Τ Σ=78 ;Τ∼0.5 Σ Σ=Τ=60 ;Τ=12 Σ=80
R.M. Koerner Ceramic Bulletin Vol. 52, No. 7 1973
property.
the only state variable.
24
25
Data courtesy of Steve Galen
Loading History Triaxial Testing SR=Stress Ratio=
axial radial
Dibasic Calcium Phosphate (A-Tab) d = 180 µm
26
Normal Strength SN Transverse Strength ST
Data courtesy of Steve Galen
The same sample after die compaction shows the different strength in transverse direction and normal direction. Anisotropy!
27
Drucker-Prager Cap
28
P.Ponte Castaneda
Continuum model with isolated pores; Takes into account the evolution of the porosity and the development of anisotropy due to change in the shape and the orientation of the voids during deformation. Micromechanics model with discrete particles; An internal state variable (B tensor) is used to describe the evolution of anisotropy under general loading. Anisotropic constitutive model Anisotropic constitutive model Micromechanics model Micromechanics model
N.A. Fleck, J. Mech. Phys. Solids 43 (1995) 1409-1431 M.Kailasam, N. Aravas, P. Ponte Castaneda CMES, Vol. 1, pp. 105-118 2000
29
N.A. Fleck, J. Mech. Phys. Solids 43 (1995) 1409-1431
Macro plastic strain Micro velocity field
ij
ij
j ij i
30
Anisotropic factor ----“B” Tensor : (i) The distribution of contact area; (ii) The number of contacts per unit surface area of particle; (iii)The hardness of each contact.
] cos ) sin (sin ) cos (sin [ ) 1 ( 4 1
2 2 2 zz yy xx zz yy xx j i ij
E E E E E E D D D n n B & & & & & & + + + + − − = φ θ φ θ φ
Hydrostatic Compaction:
) 1 ( 12 1 D D D n n B
j i ij
− − =
Constant! Die Compaction:
φ
2 0 cos
) 1 ( 4 1 D D D n n B
j i ij
− − =
31
2-D Axisymmetric Put a perturbation, calculate the differential
E W & & ∆ ∆ = ∑
Macroscopic stress may be calculated by differentiation of plastic dissipation with respect to plastic strain rate .
ij
Σ
W &
ij
E &
ij ij
E W & & ∂ ∂ = Σ
0.5 1 1.5 2
0.5 1 1.5
Initial loading point
m
∑
∑
XX ZZ
E E H 2 + = & & ) ( 3 2
XX ZZ
E E E & & & − =
Plastic strain rate
H W
m
& & ∂ ∂ = ∑ E W & & ∂ ∂ = ∑
Macroscopic stress
Die Compaction Hydrostatic Compaction
32
leads to overestimate of loads;
not be implemented into FEM)
which is constant ratio and does not vary with relative density. However, experiments show opposite trend and vary with relative density.
Normal Transverse
33
34
P
P is a material point surrounded by a material neighborhood. (macro element)
E1 E2 E3 Cracks Grain boundaries Voids Inclusions
Possible microstate of an RVE for material neighborhood of P
35
The local state is represented by the average shapes and orientations of voids All the voids initially have the same shape and orientation and distributed randomly in a elastic-plastic matrix Under finite plastic deformation, the voids remain ellipsoidal but change their volume, shape and orientation with the “local” macroscopic deformation The size of the voids is assumed to be much smaller than the scale of variation of the macroscopic fields
36
p e
is the effective elastic compliance tensor
is the spin of the voids (antisymmetric tensor)
f is porosity and Q is a microstructure tensor
) ( s I L Q − =
) 3 ( ) 2 ( ) 1 ( 2 1
X1 X2 X3
a b c
1 1
1
−
37
The plastic behavior described by the macroscopic potential is fully compressible. Φ ~
2
y
The effective yield function can be written:
y
σ
is the yield strength in tension of the matrix material.
m ~
corresponds to an appropriately normalized effective viscous compliance tensor.
N D p Λ = & σ ∂ Φ ∂ = N Λ &
is the plastic multiplier, larger than zero.
38
When the porous material deforms, the state variables evolve and, in turn, influence the response of the material.
) , ( ) 1 ( ) 1 ( s h N f D f f
kk p kk
σ Λ ≡ − Λ = − = & & &
) , ( ) (
1 ' 11 ' 33 1 1
s h D D w w
p p
σ Λ = − = & & ) , ( ) (
2 ' 22 ' 33 2 2
s h D D w w
p p
σ Λ = − = & &
) 3 , 2 , 1 (
) ( ) (
= = i n n
i i
ω &
M.Kailasam, N. Aravas, P. Ponte Castaneda CMES, Vol. 1, pp. 105-118 2000
39
Limitations: No contact area and coordination number evolution Symmetric yield surface, not appropriate for “powder” Limitations: Affine motion Stage II Compaction (RD>0.9) Stage I Compaction (RD<0.9) Ellipsoid voids with shape and
Contact area and coordination number “s” tensor “B” tensor Matrix with voids inside Interaction of particles
40
A versatile version of the Drucker-Prager model was implemented in VUMAT of ABAQUS/Explicit. State of the art models of compaction predict densification but not post processing properties because of
Fleck’s and P&A models were reviewed to check if they can address the weakness of Drucker-Prager model
Affine motion assumption; Cannot address low triaxialites cases; Predicts wrong anisotropy…
No contact area and coordination number evolution; Symmetric yield surface, not appropriate for “powder”
41
FE Model FE Model Micromechanical Model Micromechanical Model Continuum mechanics Model Continuum mechanics Model
Stage I compaction Take into account of the anisotropy in microscopic level (“B” Tensor) Modify the assumption of “affine motion” Combine micromechanical model and continuum mechanics model Develop new model and implement it to VUMAT Study the modes of fracture during diametrical compaction of tablet with new model